Shear confusion: Identification of the appropriate equivalent strain in simple shear using the logarithmic strain measure

Abstract

There is significant confusion surrounding the appropriateness of the logarithmic (Hencky) strain measure to describe simple shear deformation for finite strain. In simple shear loading of plastically deforming materials, the principal stress and strain directions do not remain coaxial which has led to conflicting derivations of the equivalent strain throughout the literature. The source of this confusion is attributed to a misapplication of the formulas for the von Mises equivalent strain and its increment that are only valid for proportional coaxial loading. In this work, a closed-form solution for the work-conjugate equivalent strain for an arbitrary yield function was derived for simple shear loading that is readily amenable to experimental characterization and is entirely consistent with the logarithmic strain measure. An analytical stress and strain integration of an elastic-plastic material was performed using the logarithmic objective rate to demonstrate that the stress, logarithmic strain, and principal directions are correctly determined within a hypo-elastic-plastic framework to finite strains. It was demonstrated that the integrated equivalent plastic strain is work-conjugate and the logarithmic strain measure is appropriate for finite simple shear. A review of the recent experimental literature for shear characterization has found that the misapplication of the von Mises equivalent strain formula for coaxial loading in simple shear loading is pervasive. The coaxial effective strain formula is the default measure in commercial digital image correlation (DIC) software and may significantly underestimate the equivalent strain in the simple shear loading condition. If the major principal strain in a simple shear test is lower than 50%, the error between the coaxial and work-conjugate equivalent strains is negligible. Otherwise, the error grows in a hyperbolic manner. Within the context of a finite element simulation of a simple shear test, if a hypo-elastic-plastic formulation is employed as in most commercial codes, the equivalent strain will be correctly computed from work conjugacy but the cumulative and principal logarithmic strain tensors will be incorrect at finite strains unless the logarithmic rate is employed.